Download Instantaneous Power

Instantaneous Power
p  vi
v  Vm cos(t   v )
i  I m cos(t  i )
ECE 201 Circuit Theory I
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Reference for t = 0
• Call t = 0 when the current is passing through
a positive maximum.
• Re-write the equations as
i  I m cos t
v  Vm cos(t  v  i )
ECE 201 Circuit Theory I
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Re-writing
p  vi
p  Vm cos(t   v  i ) I m cos t
1
1
cos  cos   cos(   )  cos(   )
2
2
  t   v   i
  t
ECE 201 Circuit Theory I
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p  Vm I m cos(t   v  i ) cos t
Vm I m
Vm I m
p
cos( v  i ) 
cos(2t   v  i )
2
2
cos(   )  cos  cos   sin  sin 
Vm I m
Vm I m
Vm I m
p
cos( v  i ) 
cos( v  i ) cos 2t 
sin( v  i ) sin 2t
2
2
2
ECE 201 Circuit Theory I
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 v  60
i  0
ECE 201 Circuit Theory I
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p is at twice the frequency of v and i
p is negative over part of the cycle, even though the elements are passive
ECE 201 Circuit Theory I
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Average and Reactive Power
p
Vm I m
V I
V I
cos( v  i )  m m cos( v  i ) cos 2t  m m sin( v  i ) sin 2t
2
2
2
Vm I m
P
cos( v  i )
2
Vm I m
Q
sin( v  i )
2
Average, or Real Power
Reactive Power
p  P  P cos 2t  Q sin 2t
ECE 201 Circuit Theory I
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Power for Purely Resistive Circuits
• For a purely resistive circuit, the voltage and
current are in phase.
 v  i
Vm I m
Vm I m
P
cos( v  i ) 
2
2
Vm I m
Q
sin( v  i )  0
2
p  P  P cos 2t  Q sin 2t  P  P cos 2t
ECE 201 Circuit Theory I
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Plot of the instantaneous Real Power
ECE 201 Circuit Theory I
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Power for Purely Inductive Circuits
• Voltage and Current are out of phase by 90°.
• The current lags the voltage by 90°.
i   v  90
 v  i  90
Vm I m
P
cos(90)  0
2
Vm I m
Vm I m
Q
sin(90) 
2
2
p  P  P cos 2t  Q sin 2t  Q sin 2t
ECE 201 Circuit Theory I
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Plot of Reactive Power (vars)
ECE 201 Circuit Theory I
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Power for Purely Capacitive Circuits
• Voltage and Current out of phase by 90°
• The current leads the voltage by 90°.
i   v  90
 v  i  90
Vm I m
P
cos(90)  0
2
Vm I m
Vm I m
Q
sin(90)  
2
2
p  P  P cos 2t  Q sin 2t  Q sin 2t
ECE 201 Circuit Theory I
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Plot of Reactive Power (vars)
ECE 201 Circuit Theory I
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Power Factor
• Power factor angle
 v  i
• Power factor
pf  cos(v  i )
• Reactive factor
rf  sin(v  i )
ECE 201 Circuit Theory I
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Power Factor (continued)
• Lagging power factor
– Current Lags the Voltage
– Inductive Load
• Leading power factor
– Current Leads the Voltage
– Capacitive Load
ECE 201 Circuit Theory I
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